Energy-Based Unmixing Method for Low Background Concentration Oil Spills at Sea

Abstract

Marine oil spills have caused severe environmental pollution with long-term toxic effects on marine ecosystems and coastal habitants. Hyperspectral remote sensing is currently used in efforts to respond to oil spills. Spectral unmixing plays a key role in hyperspectral imaging because of its ability to extract accurate fractional abundances of constituent materials from spectrums collected by sensors. However, multiple oil-propagating processes provide different mixing states of oil and water, thereby involving complicated, nonlinear mixing effects between in-depth elements in water, especially those with a low concentration. Therefore, an accurate inversion of material abundance remains a challenging yet fundamental task. This study proposes an unmixing method with normalizers in a combined polynomial and sine model to resolve overfitting problems. An energy information-based wavelet package scheme effectively highlights the latent information of the concerned material. Experimental analyses of synthetic and real data indicate that the proposed method shows superior unmixing performance, especially in delivering more accurate abundance estimations of different background oil concentration levels as low as a fractional abundance of ${10}^{-5}$, and can be used for long-term monitoring of oil propagation.

1. Introduction

With an ever-increasing demand for oil and its petroleum products, oil contamination resulting from production and transportation has become a major pollutant in the world. Furthermore, the largest and most damaging pollution events occur usually from the spills of disabled tankers, drill platforms, or broken pipelines at sea. In addition to causing obvious economic loss and immediate detrimental effects on the surrounding environment, they also possess a long-term toxicity to fauna and flora. Moreover, this soon poses significant, far-reaching harm to marine ecosystems and coastal habitants [1]. Such adverse impacts of marine oil spills have been documented in various aspects [2,3,4], even years after the accidents [5,6].

Hyperspectral remote sensing incontrovertibly plays an important role to clean oil spills and provide a fast response to oil spill accidents [7,8,9,10,11]; however, oil spills cannot be completely removed, and long-term monitoring of the oil propagation is necessary. Due to spatial resolution limits, the spectral response collected by hyperspectral sensors is a mixture of materials in the instantaneous field of view. Hyperspectral unmixing aims to separate the target pixel spectrum into a set of constituent spectral signatures (endmembers) and their fractional abundances; therefore, it is crucial for quantitative applications [9,12,13]. An accurate inversion of oil abundance helps to monitor the quantity and distribution of oil slicks. However, the water–oil mixture can be extremely complex due to oil propagation accompanied by a series of processes involving physical and chemical effects such as water-in-oil emulsification, dissolution, dispersion, sinking, and releasing of the submerged oil [5,14]. This results in a layer of oil and water mixture in different states at the sea surface and subsurface, possibly with low background concentrations of oil over a large area (Figure 1). Nevertheless, research on quantitative analysis of such situations is rare.

Methods to unmix spectral information are at the core of hyperspectral applications; furthermore, improving generic mixing models is fundamental to achieving this. Considering that, the endmember spectra are obtained from a spectral library of pure pixels available a priori or derived using extraction algorithms from image pixels [15,16,17], fractional abundances can be obtained by inverting a linear mixing model (LMM). LMM follows the assumption that the scene is flat and incident light interacts with only one constituent material at a macroscopic scale. However, LMM can be inappropriate if constituent materials interact and interfere with each other when building the overall pixel spectral signatures [18,19].

Nonlinear mixture is extensively considered for two kinds of scenarios. One typical nonlinear mixture is the intimate mixture where photonic interactions occur at a microscopic scale (e.g., imaged scenes composed of sand or mineral mixtures). Among all the intimate mixture models, the most popular approach is probably those of Hapke [18,19,20,21]; this is because they provide the most rigorous description available of light with particulate media based on physical analysis. Other popular models include the discrete-dipole approximation method [22] and the Shkuratov model [23]. However, they strongly depend on parameters inherent to the experiment and hence require near perfect knowledge of the material properties and viewing geometry, making them challenging to implement. Conversely, multilayer mixture is another typical nonlinear mixture where light has the possibility to interact with different endmembers at different depths in the macroscopic scale (e.g., trees and urban scenes). Bilinear mixing models play a crucial role in approximating the nonlinearity of layered interactions [24,25,26], which are widely used in nonlinear hyperspectral unmixing [27,28]. Among them, the linear–quadratic mixing model (LQM) derived from radiative transfer theory is justified by physical analysis, and its effectiveness can be verified in urban scenes [24].

Polynomials are thus a reliable choice for approximating nonlinear functions among endmembers in a mixture [29,30]. However, bilinear polynomials are limited to characterize interactions between only two endmembers; interplay among multiple endmembers that occurs at higher order nonlinearity is not reflected in this case. A p-linear mixing model (pLMM) is proposed to resolve this issue by involving all the effects into a p-order polynomial [31,32,33] where p is an integer and p ≥ 2. Here, a compact structure balancing computational cost and unmixing accuracy is employed. The model delivers an improved characterization of multi-interplays, especially in geometrically complex urban scenes. Further, the reconstruction performance is improved with an increased p values and stays stable. The improvement is limited in cases where the inherent polynomial nonlinearity order p of the scene is comparatively small. The limitation is emphasized in situations, such as oil spills in water, where both the macroscopic geometrical structure and constituent materials are simple. The oil spill scenes show small p values [29], and the mixture delivers widespread nonlinearity with more detailed information to be further described.

A polynomial and sine model (PSM) is proposed to characterize extensive information for a thorough description of mixed materials, thereby aiming to further enhance the reconstruction accuracy at a fixed p value [34]. Analyses of oil spill imageries have confirmed the effectiveness of the PSM. However, previous work from us focused on relatively large concentrations of oil around a spill site using PSM. Considering the different processes of spreading oil at sea, the reflectance collected in the oil spill scene is significantly impacted by scatterings derived from mixtures of oil and water in different mixing states and bathymetric interplay between the in-depth elements of water bodies at a microscopic scale. According to the mixture complexity, nonlinearity orders of PSM should be adjusted to be large, which may result in overfitting. The normalization method has been proven a powerful method to control overfitting in complex models [35,36]. Therefore, proper normalizers might be a possible solution for PSM. Simultaneously, as oil spill propagates in water for a comparatively longer time, the concentration becomes lower. The spectral signatures of low oil concentration contributing to the target pixel tend to be covered by the signatures from surrounding environment within hundreds of bands. Similar situations of inverting small abundances of constituent materials have been discussed in experiments in [25], where the estimated fractions were much smaller than the real values. However, investigations for estimating the imbalanced fractional distribution of constituent materials are rare.

To resolve these problems, this study proposes a new nonlinear unmixing method based on PSM. The proposed method integrates proper normalizers and highlights specific information brought by low concentration based on energy distributions. It aims to deliver fractional abundances of concerned materials at a higher accuracy; it can be applied to the images of oil spills at sea. This study focuses on the mixture of oil and water at the sea surface and subsurface, which can be considered as a layer of solution formed by breaking waves and other propagating conditions. The unmixing performance of the proposed model has been compared and verified based on both synthetic and real data. This method can be used to provide assistance in monitoring oil propagation after accidents and designing response operations.

2. Related Works

The LMM is not sufficient when considering multiple scatterings and interactions between endmembers. Recently, research efforts have been devoted to design nonlinear models [12]. Among all nonlinear models, polynomial is the primary choice for approximating nonlinear effects. Meanwhile, the LQM is a bilinear model derived from physical analysis, thoroughly characterizing the nonlinear interactions between two constituent materials [24]. The analytical expression for the l-th pixel in a given scene is as follows:

$$y_l={\displaystyle \sum _{r=1}^R{\alpha }_{rl}{m}_r}+{\displaystyle \sum _{i=1}^{R-1}{\displaystyle \sum _{j=i}^R{\beta }_{ijl}{m}_i\odot m_j}},$$

(1)

where $y_l={\left[y_{l_n}\right]}_{n=1,\cdots ,N}$ and, $y_{l_n}\in \mathbf{R}$ is the N-band spectral signature of the l-th pixel. Moreover, R is the number of endmembers; $m_r={\left[m_{r_n}\right]}_{n=1,\cdots ,N}$ is the spectral signature of the r-th endmember, ${\alpha }_{rl}$ is the abundance of the r-th endmember in the l-th pixel, and $0\le {\beta }_{ijl}\le 1$. Finally, $m_i\odot m_j={\left[m_{i_n}{m}_{j_n}\right]}_{n=1,\cdots ,N}$, the product form accounts for interactions between $m_i$ and $m_j$ [24].

In the LQM, abundances are extracted from linear coefficients as in LMM; however, nonlinear parts are completely ignored in fractional estimation. The method of partitioning nonlinear spectral mixture analysis (pNSMA) is another bilinear model providing fractional abundances $f_r(r=1,2,\cdots ,R)$, and is estimated by the redistribution of nonlinear fractions as follows [25]:

$$f_r=f_r^{(1)}+{\tau }_r{\displaystyle \sum _{t=1}^n{f}_{r,t}^{(2)}},$$

(2)

where $f_r{}^{(1)}$ is the fraction of the first order interaction of r-th endmember and $f_{r,t}{}^{(2)}$ de scribes the fraction of the t-th second order mixture effects involving the r-th endmember (n second order mixture effects). ${\tau }_r$ is the pixel-specific distribution factor for the r-th endmember.

These bilinear models can only describe the interaction between two endmembers according to the second order polynomial structure. To overcome this limitation and to include all multiple-element and multiple-scattering effects, a compact p-order (p ≥ 2) polynomial, defined as pLMM, is proposed [31,32,33]; it is defined as follows:

$$y_l={\displaystyle \sum _{r=1}^R{\alpha }_{rl}{m}_r}+{\displaystyle \sum _{k=2}^p{\displaystyle \sum _{r=1}^R{\beta }_{ikl}{m}_r{}^k}},$$

(3)

where ${\alpha }_{rl}\ge 0,{\beta }_{rkl}\ge 0$ and ${\displaystyle \sum _r{\alpha }_{rl}}+{\displaystyle \sum _{r,k}{\beta }_{rkl}}=1$. The model can be regarded as a linear expression of $m_r$ and $m_r{}^k$, making an efficient unmixing with low computational complexity. The fractional abundance estimation of $m_r{}^k$ is derived considering the parts containing $m_r$ in Equation (3) as a whole; hence, a polytope decomposition method (POD) has been proposed [32]. Experimental results [32] show excellent performance compared to other models because of the high-order nonlinearity description, and it is applicable to geometrically complex urban scenes. Furthermore, analyses of real data show that the reconstruction error (RE) decreases with increasing p values and reaches an optimum. However, when p is fixed, an approximating function choice other than polynomial must be considered to further enhance the reconstruction accuracy.

To detail the complexity of nonlinear effects, sine terms are introduced as an additional choice. The analytical expression of the combined PSM can be summarized as follows [34]:

$$y_l={\displaystyle \sum _{k=1}^p{\displaystyle \sum _{r=1}^R{\alpha }_{rkl}{m}_r^k}}+{\displaystyle \sum _{k^{\prime }=1}^{p^{\prime }}{\displaystyle \sum _{r=1}^R{\beta }_{r{k}^{\prime }l}\sin (k^{\prime }T{m}_r)}},$$

(4)

where $0\le {\alpha }_{rkl},{\beta }_{r{k}^{\prime }l}\le 1$ and ${\displaystyle \sum _{r,k}{\alpha }_{rkl}}+{\displaystyle \sum _{r,k^{\prime }}{\beta }_{r{k}^{\prime }l}}=1$. T is related to the period.

Considering the contribution of nonlinear parts as important supplementary information while avoiding micro-contributions covered by large values, the abundance estimation for an endmember $m_r$ is derived from all parts including it ($m_r$), and is calculated term by term as follows:

$${\overline{a}}_r={\displaystyle \sum _{k=1}^p{\alpha }_{rkl}{\Vert m_r^k\Vert }_q}/{\Vert m_r^{}\Vert }_q+{\displaystyle \sum _{k^{\prime }=1}^{p^{\prime }}{\beta }_{r{k}^{\prime }l}{\Vert \sin (k^{\prime }T{m}_r)\Vert }_q}/{\Vert m_r\Vert }_q,$$

(5)

where the q-norm ${\Vert x\Vert }_q$ for $x=(x_1,\cdots ,x_N)$ is ${\displaystyle \sum _{i=1}^n{({\left|x_i\right|}^q)}^{\frac{1}{q}}}$. The final estimated fractional abundance is defined as follows:

$${\widehat{a}}_r=\frac{\overline{a_r}}{{\displaystyle \sum _{j=1}^R\overline{a_j}}},$$

(6)

therefore, ${\widehat{a}}_r(r=1,\cdots ,R)$ fulfills the sum-to-one and non-negativity constraints.

The model comprising polynomial and sine functions is a linear expression of $m_r{}^k$($m_r$ with an exponent of k)and $\sin (k^{\prime }T{m}_r)$ (sine expression of $m_r$). The polynomial part of the function manages the main body of interactions, whereas the sine portion manages the details. Taking advantage of both parts, the PSM has delivered a better understanding of hyperspectral images, especially with respect to oil spills at sea.

3. Methods

3.1. Normalized PSM

In practical applications of hyperspectral unmixing, the scenes can be very complex with multiple interactions and interference. To accurately quantify complex physical and chemical compositions, nonlinearity orders p and p’ should be large, which in turn give rise to overfitting issues. Therefore, normalization methods are considered in PSM for efficiently controlling overfitting in complex models [35,36]. First, PSM can be considered as a linear expression as follows:

$${\mathit{y}}_l=\mathit{M}\cdot {\mathit{\theta }}_l+{\mathit{\eta }}_l,$$

(7)

where

$$\mathit{M}=[{\mathit{m}}_1,\cdots {\mathit{m}}_R,\cdots ,{\mathit{m}}_1^p,\cdots {\mathit{m}}_{\mathit{R}}^{\mathit{p}},\cdots ,\sin (T{\mathit{m}}_1),\cdots \sin (T{\mathit{m}}_R),\cdots ,\sin (p^{\prime }T{\mathit{m}}_1),\cdots \sin (p^{\prime }T{\mathit{m}}_R)],$$

(8)

$${\mathit{\theta }}_{\mathit{l}}=[{\mathit{\alpha }}_{1,1,l},\cdots {\mathit{\alpha }}_{R,1,l},\cdots ,{\mathit{\alpha }}_{1,p,l},\cdots {\mathit{\alpha }}_{R,p,l},\cdots ,{\mathit{\beta }}_{1,1,l},\cdots {\mathit{\beta }}_{R,1,l},\cdots ,{\mathit{\beta }}_{1,p^{\prime },l},\cdots {\mathit{\beta }}_{R,p^{\prime },l}],$$

(9)

and ${\eta }_l$ is the noise of the observed data. On the assumption of Gaussian properties of noise,

$$y_l\sim N(M\cdot {\theta }_l,{\sigma }^{-1}I),$$

(10)

the probability density function (PDF) is defined as follows:

$$p({\mathit{y}}_l|{\mathit{\theta }}_l,\sigma )={\left(\frac{\sigma }{2\pi }\right)}^{\frac{N}{2}}\exp \{-\frac{\sigma }{2}{\Vert {\mathit{y}}_l-\mathit{M}\cdot {\mathit{\theta }}_l\Vert }_2^2\},$$

(11)

where ${\sigma }^{-1}I$ is the covariance matrix. Assuming that $\sigma$ is known, the parameters can be estimated by means of maximum likelihood estimation as follows:

$$\begin{array}{cc}\hfill {\hat{\mathit{\theta }}}_l& =\arg \underset{{\mathit{\theta }}_l}{\max }\ln p({\mathit{y}}_l|{\mathit{\theta }}_l,\sigma )\hfill \\ & =\arg \underset{{\mathit{\theta }}_l}{\max }\{-\frac{\sigma }{2}{\Vert {\mathit{y}}_l-\mathit{M}\cdot {\mathit{\theta }}_l\Vert }_2^2+\frac{N}{2}\ln \sigma -\frac{N}{2}\ln (2\pi )\}\hfill \\ & =\arg \underset{{\mathit{\theta }}_l}{\max }\{-\frac{\sigma }{2}{\Vert {\mathit{y}}_l-\mathit{M}\cdot {\mathit{\theta }}_l\Vert }_2^2\}\hfill \\ & =\arg \underset{{\mathit{\theta }}_l}{\min }\left\{\frac{\sigma }{2}{\Vert {\mathit{y}}_l-\mathit{M}\cdot {\mathit{\theta }}_l\Vert }_2^2\right\}.\hfill \end{array}$$

(12)

For a specific mixture, a priori information of parameters can be estimated. In this study, a priori distribution is considered using Bayesian method with the assumption that

$${\theta }_l\sim N({\overline{\theta }}_l,{\lambda }^{-1}I),$$

(13)

where ${\overline{\theta }}_l$ is the a priori average parameter value and ${\lambda }^{-1}I$ is the a priori covariance of the parameter vector. The PDF of ${\theta }_l$ is as follows:

$$p({\mathit{\theta }}_l|\lambda )={\left(\frac{\lambda }{2\pi }\right)}^{\frac{(p+p^{\prime })R}{2}}\exp \{-\frac{\lambda }{2}{\Vert {\mathit{\theta }}_l-{\overline{\mathit{\theta }}}_l\Vert }_2^2\}.$$

(14)

According to the Bayes theorem,

$$p({\theta }_l|y_l,M,\sigma ,\lambda )\propto p(y_l|{\theta }_l,\sigma )p({\theta }_l|\lambda ).$$

(15)

Furthermore, applying maximum likelihood estimation to the PDF logarithm gives the following expression:

$${\hat{\mathit{\theta }}}_l=\arg \underset{{\mathit{\theta }}_l}{\min }\{\frac{\sigma }{2}{\Vert {\mathit{y}}_l-\mathit{M}\cdot {\mathit{\theta }}_l\Vert }_2^2+\frac{\lambda }{2}{\Vert {\mathit{\theta }}_l-{\overline{\mathit{\theta }}}_l\Vert }_2^2\},$$

(16)

Dividing Equation (16) by $\sigma$, it can also be expressed in an equivalence form as follows:

$${\hat{\mathit{\theta }}}_l=\arg \underset{{\mathit{\theta }}_l}{\min }\{\frac{1}{2}{\Vert {\mathit{y}}_l-\mathit{M}\cdot {\mathit{\theta }}_l\Vert }_2^2+\frac{\lambda }{2}{\Vert {\mathit{\theta }}_l-{\overline{\mathit{\theta }}}_l\Vert }_2^2\}.$$

(17)

It should be noted that the optimization in Equation (17) is equivalent to a normalized PSM with the normalizer (the second part) being added to the RE of the model (the first part) as a first choice. Furthermore, to balance the objective function for algorithm efficiency, another normalizer is added as an extension due to the small-value parameter constraints. Thus, the final normalized PSM can be described as follows:

$${\hat{\mathit{\theta }}}_l=\arg \underset{{\mathit{\theta }}_l}{\min }\{\frac{1}{2}{\Vert {\mathit{y}}_l-\mathit{M}\cdot {\mathit{\theta }}_l\Vert }_2^2+\frac{\lambda }{2}{\Vert {\mathit{\theta }}_l-{\overline{\mathit{\theta }}}_l\Vert }_2^2+\frac{\mu }{2}{\Vert {\mathit{\theta }}_l\Vert }_2^2\},$$

(18)

where $\mu$ is set to be a nonnegative small-value and $\lambda >\mu$.

3.2. Energy-Based Wavelet Package Decomposition Method

Even though successful mixing models are introduced for the better understanding of complex mixtures, quantitative unmixing remains a challenge for scenarios containing large amount of detailed information. The effect is emphasized when the existence of certain materials shows a low concentration, or even when these act as latent information covered in a massive data delivered by a hyperspectral cube.

Fourier and wavelet transformations possess the advantages of highlighting low- and high-frequency information. The wavelet package decomposition method is able to manage detailed information and content estimation [37,38]. A low-concentrated constituent element in spectral information will be scattered throughout the entire spectrum range. However, it carries a low energy level. Therefore, apart from low- or high-frequency choices, we introduce a wavelet package method based on PSM for combining high-frequency and low-frequency information, where the contributions are dependent on their individual energy proportion. Since the wavelet package transformation follows a linear property, the linearity of Equation (7) holds for any node k at any level L in the wavelet package transform tree as follows:

$${\psi }_{k,L}({\mathit{y}}_l)={\psi }_{k,L}(\mathit{M})\cdot {\mathit{\theta }}_{k,l}+{\psi }_{k,L}({\mathit{\eta }}_l),$$

(19)

where ${\psi }_{k,L}(x)(k=0,1,\cdots ,2^L-1)$ are the wavelet coefficients of $x$, and

$$\begin{array}{cc}\hfill {\psi }_{k,L}(\mathit{M})=& [{\psi }_{k,L}({\mathit{m}}_1),\cdots {\psi }_{k,L}({\mathit{m}}_R),\cdots ,{\psi }_{k,L}({\mathit{m}}_1^p),\cdots {\psi }_{k,L}({\mathit{m}}_{\mathit{R}}^{\mathit{p}}),\cdots ,\hfill \\ & {\psi }_{k,L}(\sin (T{\mathit{m}}_1)),\cdots {\psi }_{k,L}(\sin (T{\mathit{m}}_R)),\cdots ,{\psi }_{k,L}(\sin (p^{\prime }T{\mathit{m}}_1)),\cdots {\psi }_{k,L}(\sin (p^{\prime }T{\mathit{m}}_R))],\hfill \end{array}$$

(20)

Then, parameters ${\theta }_{k,l}$ are estimated for each node. Summing all the obtained parameters with the energy of each node as the corresponding weighting factor, the estimated parameter vector is designed as follows:

$${\hat{\mathit{\theta }}}_l=E_{L,0}{\hat{\mathit{\theta }}}_{0,l}+E_{L,1}{\hat{\mathit{\theta }}}_{1,l}+\cdots +E_{L,2^L-1}{\hat{\mathit{\theta }}}_{2^L-1,l},$$

(21)

where the wavelet package energy spectrum at L level is defined as follows:

$$E_L=[E_{L,0},E_{L,1},\cdots ,E_{L,2^L-1}].$$

(22)

Since the specific information regarding low concentration levels are highlighted by means of the proposed scheme, accurate abundances can be expected in the unmixing process.

3.3. ENPSM Unmixing Method

An energy based normalized polynomial and sine model (ENPSM) has been proposed. Considering specific a priori information, two normalizers have been added to the PSM. Meanwhile, energy information has found expression in abundance estimation.

First, the wavelet package transformation to pixels and the extended linear endmember parts as described in Equation (8) is implemented. Then, sets of parameters are estimated on each node of the same level by means of PSM with normalizers in Equation (18). Considering the specific situation of low concentration, the concerned material covers only a relatively small fraction (particularly even when approaching trace level), in which the a priori average of the relevant parameters ${\overline{\theta }}_{lc}$ can be set to zero. Moreover, ${\overline{\theta }}_l$ comprises ${\overline{\theta }}_{lc}$ and the a priori parameters averaging the rest of the elements, denoted as ${\overline{\theta }}_{lc}{}^c$. The coupling of two parts satisfying the sum to one constrain can be controlled only by considering any one of them. Denote ${\theta }_{lc}$ as the parameters of the concerned material. The first normalizers of Equation (18) can be simplified as follows:

$$\frac{\lambda }{2}{\Vert {\mathit{\theta }}_{lc}-{\overline{\mathit{\theta }}}_{lc}\Vert }_2^2=\frac{\lambda }{2}{\Vert {\mathit{\theta }}_{lc}\Vert }_2^2,$$

(23)

which is consistent with the fact that relatively small effects of the concerned materials contribute to the total nonlinear interaction. Accordingly, the objective function with image pixels and endmembers replaced by those found in Equations (19) and (20) can be summarized as follows:

$${\hat{\mathit{\theta }}}_{k,l}=\arg \underset{{\mathit{\theta }}_l}{\min }\{\frac{1}{2}{\Vert {\psi }_{k,L}({\mathit{y}}_l)-{\psi }_{k,L}(\mathit{M})\cdot {\mathit{\theta }}_{k,l}\Vert }_2^2+\frac{\lambda }{2}{\Vert {\mathit{\theta }}_{k,lc}\Vert }_2^2+\frac{\mu }{2}{\Vert {\mathit{\theta }}_{k,l}\Vert }_2^2\}(k=0,1,\cdots ,2^L-1).$$

(24)

Finally, the parameters are estimated based on energy distribution as given in Equation (21); hence, fractional abundances can be derived from Equations (5) and (6).

4. Experiments and Results

4.1. Experimental Data

The proposed method was validated by testing on synthetic and real hyperspectral data. First, the proposed architecture performance was tested over five synthetic images with similar size (50 × 50 pixels) generated by the LQM with the endmembers of crude oil and seawater. The spectral signatures of endmembers were collected using 2151 bands between 350 and 2500 nm by the spectrometer of the analytical spectral devices (ASD) (Figure 2). The abundance distribution of oil in the first image was designed as shown in Figure 3a. In the other four synthetic images, the abundances of oil were derived from the abundance matrix of the first image multiplied by ${10}^{-1}$, ${10}^{-2}$, ${10}^{-3}$, and ${10}^{-4}$, respectively. Endmembers were also mixed according to the Hapke’s model for another five synthetic images using the abundances designed earlier.

Given the seriousness of environmental pollution caused by oil spills, quick and effective response to emergencies, as well as the assessment of the disposal effectiveness, has become a concern for the authorities in coastal countries. After an early stage detection of oil spill accidents, long-term propagation tracking and monitoring their extent have proven to be of vital importance in pollution response. The spill in the Gulf of Mexico in 2010, known as the Deepwater Horizon (DWH) explosion, has caused serious environmental contamination and has been widely reported in the media. Hyperspectral images of the oil spill accident were collected by the airborne visible infrared imaging spectrometer (AVIRIS) sensor using 224 spectral bands in the range of 380–2500 nm. The data are available in calibrated at-sensor radiances from the NASA Jet Propulsion Laboratory. Hyperspectral data recorded on 13 May and 9 July 2010, processed by atmospheric correction module, were used to test the proposed method. Two groups of images are presented in Figure 4 and Figure 5. The main parameters are listed in

Table 1. Main parameters of images in Figure 4 and Figure 5.

Name		Attribute
Sensor		AVIRIS
Spectral range (nm)		380–2500
Number of bands		224
Date of acquisition		13 May 2010
Size (pixels)	Figure 4b	86 × 89
	Figure 4c	114 × 123
	Figure 4d	108 × 109
Date of acquisition		9 July 2010
Size (pixels)	Figure 5a	49 × 52
	Figure 5b	57 × 65
	Figure 5c	31 × 31

.

It is worth noting that, despite a low-oil chemical signal, significant biological effects of spills were detected in the water column after the DWH accident [2]. The situation remained the same even when there were only trace concentrations of oil components on water surface. Therefore, a simulated oil spill imagery of 50 × 50 pixels was constructed. The spectral signatures of constituent materials were obtained from pure pixels in Figure 4. Meanwhile, the oil distribution was simulated as shown in Figure 6. In order to integrate different content levels of oil into one image, the abundance map was divided into five parts: the original abundance, and the original abundance multiplied by ${10}^{-1}$, ${10}^{-2}$, ${10}^{-3}$, and ${10}^{-4}$. The final abundance distributions are shown in Figure 7a.

4.2. Validation Approach

The proposed ENPSM was tested and compared with classic approximating models such as pNSMA, LQM, pLMM with POD, and PSM. When inverting the analytical expressions of the five aforementioned methods, the same scheme was employed as fully constrained least squares to make an accurate comparison of the unmixing performances [39]. The reconstruction performance can be quantitatively represented using the RE (for p pixels with N bands) as follows:

$$RE=\sqrt{\frac{1}{PN}{\displaystyle \sum _{l=1}^N{\Vert y_l-{\widehat{y}}_l\Vert }_2^2}},$$

(25)

where ${\widehat{y}}_l$ identifies the reconstructed spectral signature and the abundance root mean square error (RMSE) is defined as follows:

$$RMSE=\sqrt{\frac{1}{P}{\displaystyle \sum _{l=1}^N{\Vert {\mathit{a}}_l-{\hat{\mathit{a}}}_l\Vert }_2^2}},$$

(26)

where ${\widehat{a}}_l$ corresponds to the estimated endmembers’ abundance distribution of the l-th pixel. In light of low concentration, the abundance estimation accuracy, reflected by errors in orders of magnitude, can be represented by taking the logarithm in RMSE (LOGRMSE) as shown below:

$$LOGRMSE=\sqrt{\frac{1}{P}{\displaystyle \sum _{l=1}^N{\Vert \log ({\mathit{a}}_l)-\log ({\hat{\mathit{a}}}_l)\Vert }_2^2}},$$

(27)

where $a_l$ and ${\widehat{a}}_l$ are only the abundances and estimated abundances of the concerned materials, respectively; furthermore, $\log (a)$ is set to ${10}^{-300}$ when $a=0$ in this study.

4.3. Synthetic Data

To quantitatively evaluate the performance of the proposed method, ten synthetic images were generated based on the LQM and Hapke’s model. The oil abundance magnitude ranged from ${10}^{-1}$ to ${10}^{-5}$. Due to the simple composition of the mixtures, the imagery of the oil spill delivers a small polynomial nonlinearity order [34]. The performance reaches the optimum when the value of p is between 2 and 3. To compare the reconstruction performance, polynomial nonlinearity orders of five unmixing methods were unified to be two. According to the discussion in [34], due to the constituent materials of oil and water, T = 1 and p’ = 1 is used for unmixing by PSM. The proposed LOGRMSE was compared as a measurement specially taking oil as the concerned material to evaluate the unmixing performance.

Table 2. Reconstruction performances: RE, RMSE, and LOGRMSE results produced by LQM, pNSMA, pLMM, PSM, and ENPSM over synthetic images generated based on Hapke’s equations and LQM using endmembers of oil and seawater. The best results are highlighted in bold.

Images	Mixture Model	Magnitude Order	Reconstruction Performance	LQM	pNSMA	2LMM + POD	PSM	ENPSM
Image 1	LQM	${10}^{-1}$	RE	0.0022	0.0022	0.0022	0.0022	0.0024
			RMSE	0.0327	0.0439	0.0327	0.0334	0.0929
			LOGRMSE	11.9321	0.3378	0.6974	0.5971	0.0802
Image 2	LQM	${10}^{-2}$	RE	0.0023	0.0022	0.0022	0.0022	0.0028
			RMSE	0.0318	0.0777	0.0318	0.0334	0.0288
			LOGRMSE	97.9863	298.5316	5.7726	5.0710	0.3350
Image 3	LQM	${10}^{-3}$	RE	0.0022	0.0022	0.0022	0.0022	0.0031
			RMSE	0.0076	0.0076	0.0076	0.0076	0.0065
			LOGRMSE	290.4378	291.6509	16.8534	40.6913	0.7006
Image 4	Hapke	${10}^{-1}$	RE	0.0022	0.0022	0.0022	0.0022	0.0027
			RMSE	0.0451	0.0604	0.0441	0.0445	0.0678
			LOGRMSE	0.0348	0.0662	0.0499	0.0838	0.0568
Image 5	Hapke	${10}^{-2}$	RE	0.0022	0.0022	0.0022	0.0159	0.0039
			RMSE	0.0574	0.0591	0.0144	0.0159	0.0190
			LOGRMSE	6.8086	6.2395	4.3820	3.9218	0.2243

shows the performance results in terms of RE, RMSE, and LOGRMSE. The RE values range over a small scale while those of ENPSM are a little larger. The results are mainly because in ENPSM the reconstructions were created using endmembers and parameters, whereas the parameters were obtained by inverting the vectors of wavelet package coefficients, and hence, they do not match each other. However, the vectors remained the same in other models and have smaller RE. The RMSE results show the magnitude of real data at a lower oil concentration (Images 2, 3, and 5 in Table 2) because the estimated abundances are much smaller than the real values with a difference in their magnitude. Accordingly, LOGRMSE can be a reliable measurement at lower oil concentrations. The ENPSM definitely shows superior performance in LOGRMSE over synthetic images generated by two kinds of mixing models (LQM and Hapke’s model). However, the other four unmixing methods lost accuracy for oil abundance at a lower concentration, either only deducing to zero or a small value of about ${10}^{-20}$. Therefore, Table 2 lists the results of only five images. Particularly, in Figure 1, which is listed in Table 2, the four points scattered at four corners of the image when unmixing with the LQM show very small values of oil abundances using a designed actual value of 0.02, as shown in Figure 3a. The recalculated LOGRMSE = 0.0414 after avoiding those four scattered points. Distortions at the four corners are reported in

Table 3. Recalculated results of LOGRMSE after avoiding the underestimated pixels from corners of figures.

Figure	Unmixing Method	Number of the Underestimated Pixels	LOGRMSE
Figure 1	LQM	4	0.0414
Figure 2	2LMM + POD	1	0.3856
Figure 2	PSM	28	0.4156
Figure 5	2LMM + POD	162	0.2507
Figure 5	PSM	182	0.2666

. Other, much larger LOGRMSE values result from more zeros in estimation and are in line with the results in [25].

The complete unmixing results of LOGRMSE using ENPSM are listed in

Table 4. LOGRMSE results produced by ENPSM over ten synthetics images generated based on LQM and Hapke’s equations.

Mixture Model	Magnitude Order	LOGRMSE	Mixture Model	Magnitude Order	LOGRMSE
LQM	${10}^{-1}$	0.0802	Hapke	${10}^{-1}$	0.0568
	${10}^{-2}$	0.3350		${10}^{-2}$	0.2243
	${10}^{-3}$	0.7006		${10}^{-3}$	0.6466
	${10}^{-4}$	0.8798		${10}^{-4}$	0.8671
	${10}^{-5}$	0.5249		${10}^{-5}$	0.9490

. It is notable that the ENPSM unmixing performance is stable when magnitude order decreases, thereby maintaining oil abundances similar to or at close order to the real one. The estimated abundance distribution also shows a clear map of the concerned material, even for a low abundance magnitude of ${10}^{-3}$ (Figure 3b), with drawbacks only appearing at four corners. However, ENPSM also limits accuracy when the magnitude is ${10}^{-6}$, and this needs to be investigated through laboratory analysis.

4.4. Oil Spill in the Gulf of Mexico

The performance of the proposed method was tested using two groups of real data collected in the DWH, and then, it was compared with LQM, pNSMA, pLMM with POD, and PSM. The endmembers of oil and seawater were derived by the minimum volume simplex analysis (MVSA) extraction algorithm [17]. It is obvious that the three images of Figure 4b–d were captured in Figure 4a at a certain distance from the main propagation route. The propagation was accompanied by processes, such as dispersion and sinking, possibly with a lower concentration.

Table 5. RE results delivered by five unmixing method over Figure 4b–d with endmembers derived by means of maximum volume simplex analysis (MVSA) extraction algorithm [17]. The best results are highlighted in bold.

Figure	LQM	pNSMA	2LMM+POD	PSM	ENPSM
Figure 4b	0.0138	0.0138	0.0138	0.0206	0.0161
Figure 4c	0.0119	0.0119	0.0119	0.0179	0.0481
Figure 4d	0.0133	0.0134	0.0133	0.0123	0.0147

reports RE performances over images in Figure 4b–d. The original images and the estimated abundance distribution maps are shown in Figure 8.

Similar unmixing performance was observed for the first four methods listed in Table 5. They proved to be effective in scenarios where the constituent materials cover balanced abundances. However, when the mixture is complicated in its chemical and physical compositions, ENPSM is able to deliver a clearer description of the effects involved in the images. It should be noted that the abundance maps derived from ENPSM are able to show accurate oil propagations from the middle right to the upper left of Figure 4b and from the right to the left in Figure 4c (Figure 8). Another group of images in Figure 5, captured near a main propagation route, were also used to test the proposed method. The RE results listed in

Table 6. RE results delivered by five unmixing method over Figure 5a–c by means of maximum volume simplex analysis (MVSA) extraction algorithm [17].

Figure	PSM	ENPSM
Figure 5a	0.0090	0.0147
Figure 5b	0.0504	0.0208
Figure 5c	0.0098	0.0176

indicate that ENPSM can also deliver a stable performance in a scenario with comparatively balanced abundances, as in Figure 4d.

4.5. Constructed Oil Spill Imagery

In order to test the proposed method in actual application, different mixtures of oil spills were simulated for lower oil concentrations using spectral signatures collected from the real data of the DWH. Five continuous orders of magnitude of oil content were integrated into one image. Applying the proposed ENPSM, the total LOGRMSE = 0.5057, indicating that the average abundance error is about half a magnitude order for the entire image. The first part of the estimated oil abundance map (Figure 7b) is very close to the original distribution (Figure 7a). The next three parts in Figure 7b are able to maintain the main shape of oil spills but have higher values than the real fractions. In the last part, shapes of oil are most obscure, though all the estimated fractions remain at the order of ${10}^{-5}$.

5. Discussion

5.1. Comparison of Three Validation Approaches

According to the unmixing results from synthetic data (Table 2), RE values of ENPSM were a little larger than those of the other four unmixing methods. This is mainly because the objective function of ENPSM was not the original RE in Equation (25), as $y_l$ and endmembers in the objective function were replaced by vectors of their wavelet package coefficients. For the images collected in the DWH, the RE results were similar to those for the synthetic data. In terms of RMSE, Equation (26) cannot reflect the actual errors between the real abundances and the estimated ones, when the extracted abundances were different from the real values in magnitude; thus, RMSE will only reflect large values under this circumstance. For example, in Figure 2, Figure 3, and Figure 5 of synthetic data, smaller oil abundances were underestimated for a certain number of pixels in LQM, pNSMA, pLMM with POD, and PSM, with the invasion result of a small value of about ${10}^{-20}$ or directly returning to zero. Therefore, RMSE is not effective at a smaller abundance of constituent materials. Conversely, LOGRMS exhibits accuracy in this situation, and can reliably compares the magnitudes of $a_l$ and ${\widehat{a}}_l$. However, in images with comparatively larger material abundances, RMSE shows a good validation performance, as can be seen in Figure 1 and Figure 4. Therefore, it is appropriate to combine RMSE and LOGRMSE to assess average errors in estimated abundances for different magnitudes.

5.2. Comparison of Five Unmixing Methods

LQM demonstrated a good unmixing performance in experimental data, as seen in Figure 1 and Figure 4, with comparatively balanced abundances in RE, RMSE, and recalculated LOGRMSE (Table 3). However, it may have low accuracy for estimated material abundances, as in the corners of Figure 1. The oil abundances of Figure 1 were designed as in Figure 3a, with most pixels in the magnitude order of ${10}^{-1}$ and four corner points equaling 0.02. In contrast, pNSMA was able to maintain the accuracy at this magnitude. pLMM and PSM showed a good unmixing performance, even in Figure 2 and Figure 5; underestimation was noted in only a certain number of pixels. However, all four of the unmixing methods were unable to estimate the fractional abundances of constituent materials when the magnitude was decreased to a value, as in Figure 3. They could only identify that the values were very small, but failed to distinguish the exact magnitude, and they even directly returned to zero for a large number of pixels. This is detrimental for the information delivered from unmixing over the images of oil spill sites after conducting clean-up operations because these results can erroneously indicate that the spill has been cleaned up already. This will result in the cutting off of subsequent work. ENPSM shows superior performance in LOGRMSE even at lower abundance magnitude in synthetic data (Table 4). From the real data of oil spills that were collected from DWH, ENPSM delivered clearer oil distribution maps. Images containing a certain number of pixels with low oil concentration (estimated abundance < 0.1) were observed in Figure 4b,c. This is based on the estimated abundances derived from the five unmixing methods (Figure 9) and coincided with the propagating locations in Figure 4a. Accordingly, ENPSM could be used to depict accurate oil propagations away from main route with low concentrations. The detailed and latent information can be highlighted using this unmixing method. Indeed, ENPSM can deliver a stable performance for different oil concentrations, including scenes with balanced constituent material abundances, as shown in the synthetic data and images in Figure 4, Figure 5, and Figure 7. Hence, by combining it with oil drift modeling studies, it can be applied to monitor and predict oil propagation [40].

5.3. Limitation

Nonlinear interactions between materials have been documented and emphasized in scenes that include water [24,32]. Light may interact with different endmembers at different depths [41]. Such interactions in water bodies cannot be neglected and contain important information regarding the constituent materials. Oil propagating at the water surface and subsurface can be considered as a layer of oil–water mixture. The estimated abundance is the volume ratio of oil in this layer. However, oil propagation is accompanied by a series of complex processes that includes physical and chemical effects. For example, in oil emulsification, oil and emulsified oil have different spectral characteristics [42]. More endmembers or image classification based on spectrum difference [42] should be considered in future work to improve the unmixing accuracy. A future work will compare and combine the proposed method with machine learning techniques to improve it, aiming to deliver more accurate information to respond to oil spills.

6. Conclusions

In this study, an unmixing scheme based on PSM was addressed. For the complicated, nonlinear effects of the constituent materials, proper normalizers are used to solve the overfitting problem, and an energy-based wavelet package method is introduced to highlight detailed information. Compared with other classic unmixing methods, the proposed ENPSM shows better reconstruction performances even when the fractional abundances of the concerned material decrease to a concentration as low as ${10}^{-5}$. The proposed algorithm shows superior unmixing performance for oil spill mixtures, especially with different concentrations. It helps to assess the effectiveness of oil cleaning after an accident and master the collection of information on oil spill spreading in sea water.

Although the scheme is not constrained to this scenario, it can be applied to analyzing the information of oil propagation to assist the long-term monitoring of and response design regarding oil spills.